Haptic Interfaces and Virtual...

Haptic Interfaces and Virtual Environments

• A haptic interface allows for a human to interact with a

computer through the sense of touch.

- The user moves a “puck” (joystick, wheel) in the workspace.

- The CPU senses this motion, and computes reaction forces

that are transmitted to the human using the motor.

- A “force feedback” system

Human

HapticDevice

actuators(muscles)

CPU sensor(encoder, tach,strain gauge)

actuator(DC motor)

sensor(skin surface)

forcepositionvelocityforce

behavior

algorithm(virtual world)

• The algorithm used by the CPU to process the position and/or

velocity measurements to produce the reaction forces is called

a “virtual environment” or “haptic world”

- virtual wall [1]

- virtual sprung mass [1]

- surgery simulator (virtual body tissue) [2]

EECS461, Lecture 8, updated October 1, 2008 1

Virtual Wall

• Consider a puck in a linear workspace:

Puck

z

virtual wall

zw

workspace limit

workspace limit

• Our goal is to create a virtual wall to prevent us from moving

the puck all the way to the right of the workspace

• Idea

- If the puck is to the left of the “virtual wall”, do nothing, so

the puck moves freely.

- If the puck is to the right of the “virtual wall”, have the

motor exert a force to push it back to the left

• To implement a virtual wall, we require

- a position measurement from an encoder

- an algorithm to compute the force, implemented on a

microprocessor

- a motor to exert the force


Virtual Spring Algorithm

• To make a virtual wall, we suppose that when the puck is to

the right of the wall, it is pulled back to the wall with a spring:

Puck

z

zw

k

• Hence we need the motor to exert a force equal to the restoring

force of a spring

F = −k(z − zw)

• Pseudocode for virtual wall:

set z_wset spring constant kwhile(1) {

read puck position z from potentiometer (encoder)if z > z_w

compute force F = k*(z - z_w)output force to DAC (set PWM duty cycle)

elsedo nothing

end}


Issues with Virtual Wall

• Rotary vs. linear workspace is analogous:

- angular rather than linear displacement

- torque rather than force

• Chatter: an “artifact” that distinguishes a computer generated

wall from an actual wall. Occurs due to combination of

- large restoring force

- slow sample time

- large quantization error

• If a tachometer is used to sense velocity, then the velocity must

be numerically integrated to obtain position displacement

• For a more detailed schematic of linear and rotary haptic

interfaces, see [1]


Virtual Sprung Mass

• Suppose we wish the puck to behave as though it had a mass

attached to it with a spring:

Puck

z

w

k

m

• If we move the puck to a new position, z, then

- the spring will exert a force F = −k(w − z) on the mass

- the force will accelerate the mass according to Newton’s

laws: −k(w − z) = mw

- the position of the mass will change acccording to

w +k

mw =

k

mz (1)

- the puck experiences a reaction force equal in magnitude

and opposite in direction to the force exerted on the mass:

Puck

z

m

w

k F FF F


Simulink Model of Spring Mass System

• The differential equation

w =k

mz −

k

mw

has a block diagram (cf. the notes on second order systems

from Lecture 7):

Σ-

z

k/m

w

k/m

.w..

w

• It is easy to show that this diagram is equivalent to one from

which the force acting on the mass, F = −k(w− z), can be

directly computed:

Σ

-

z

1/m

w

k

.w

..wF

• an appropriate SIMULINK diagram:

wwdotwddotz

4

Force3

Puck Position

2

Velocity

1

Virtual Mass Position

k

spring constant

Step

1s

Integrator1

1s

Integrator

1/m

1/virtual mass


Simulated Response

• Let us move the puck one unit to the right and hold it there.

- The mass will respond according to (1)

- A reaction force will be exerted on the puck

- This force must be countered by the user to hold the puck

stationary!

• Matlab simulation1 of mass motion, velocity, and force exerted

on the puck (assumes that the puck and mass have no physical

dimension)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−10

−8

−6

−4

−2

0

2

4

6

8

10

time, seconds

k = 10, m = 1, period of oscillation = 1.9869

mass position, wmass velocity, dw/dtpuck position, zforce on mass, F=−k(w−z)reaction force on puck, −F

• Note: the spring-mass system is an undamped harmonic

oscillator with

- frequency in radians/second, ωn =p

k/m

- frequency in Hz, fn = ωn/2π

- period in seconds, Tn = 1/fn

1Obtained from Matlab files virtual spring mass.m and oscillator force.mdl.


Algorithm for Sprung Mass

• If puck is not held constant, then the force will depend on the

puck position.

• To simulate the motion of the puck requires that we model

- the response of the puck states (position, velocity) to the

reaction force.

- the effect of the human interacting with the puck

• To implement the haptic interface,

- we measure the position of the puck.

- we do not need to simulate puck motion or human

interaction.

• Pseudocode for sprung mass:

set k, mwhile(1) {

read puck position z from potentiometer (encoder)update w according to eqn.(1)compute reaction force F = k*(w-z)output force to DAC (set PWM duty cycle)

}

• Question: How to update w?

- Must numerically integrate a second order differential

equation


State Space Form of Differential Equations

• The differential equation governing the motion of the mass is

a second order equation:

Σ

-

z

1/m

w

k

.w

..wF

• It is convenient to represent this as two first order equations.

Define state variables position and velocity:

x1 , w

x2 , w

• Define the input as the puck position, u = z, and the output

as the reaction force, y = −F = k(w − z). Then the state

differential equations are

x1 = x2

x2 = −k

mx1 +

k

mu

y = kx1 − ku


Matrix Form of State Equations

• Define the state vector

x ,

»x1

x2

–=

»w

w

–• Then the state equations can be written in matrix form as

x = Ax + Bu

y = Cx + Du

where

A ,

»0 1

− km 0

–, B ,

»0km

–, C ,

ˆk 0

˜, D , −k

• It is true in general that any n-th order differential equation

may be written as a system of n first order differential

equations, one for each of n state variables

• Example:

- Suppose that, instead of holding the puck position fixed

after an initial step change, we allow the puck to move freely

in response to the reaction force exerted by the mass.

- Need four state variables (position and velocity of both puck

and mass)


Numerical Integration of State Equations

• Consider a differential equation x = f(x, u)

• Question: How to find the value of

x(t) = x(0) +

Z t

0

f(x, u)dt

given some initial condition x(0), and the input u(t)?

• Idea: Select a time interval T , and generate an approximating

sequence x(0), x(T ), x(2T ), . . . using a difference equation

x((n + 1)T ) = x(nT ) + Tf(x(nT ), u(nT )),

with initial condition x(0) = x(0).

• The forward Euler integration algorithm

f(x,u)

T 2T 3T 4T

• The difference x((n + 1)T ) − x(nT ) is the area of one

rectangle.

• We usually abbreviate x(k) , x(kT ) and suppress the ˜

notation.


Pseudocode for Virtual Spring-Mass

• In our case, f(x, u) = Ax + Bu, where

A =

»0 1

−k/m 0

–, B =

»0

k/m

–• Hence

x(n + 1) = x(n) + T (Ax(n) + Bu(n))

= (I + TA)x(n) + TBu(n)

• x1 = position, x2 = velocity

x1(n + 1) = x1(n) + Tx2(n)

x2(n + 1) = x2(n) + T (k/m)(−x1(n) + u(n))

• Pseudocode for sprung mass:

initialize parameters k,m;initialize states x1prev = z, x2prev = 0;set update period T;while(1) {

read in puck position u = z;x1 = x1prev + T*x2prev;x2 = x2prev + T*(-(k/m)*x1prev + (k/m)*u);compute reaction force y = F = k*(x1-u)x1prev = x1; x2prev = x2;output force value

}


Difference Equations and the z-Transform

• Analogous to Laplace transform for continuous time systems

and differential equations

• Given a discrete sequence

{y(k)} = . . . , y(−2), y(−1), y(0), y(1), y(2), . . . ,

the z-transform is

Y (z) =

∞Xk=−∞

y(k)z−k

• The z-transform of the time-shifted sequence {y(k)} =

{y(k + n)} is

Y (z) = znY (z)

• The transfer function for the system with difference equation

y(k + 2) + a1y(k + 1) + a2y(k) = b1u(k + 1) + b2u(k)

is found from

Y (z) = G(z)U(z), G(z) =(b1z + b2)

(z2 + a1z + a2)


Discrete Time Simulation

• Difference equations

x1(n + 1) = x1(n) + Tx2(n)

x2(n + 1) = x2(n) + T (k/m)(−x1(n) + u(n))

• Transfer functions

X1(z) =T

z − 1X2(z)

X2(z) =T

z − 1(k/m)(−X1(z) + U(z))

• For us, x1 = w, x2 = w, u = z

• SIMULINK model2

w

wdot

wddotz

3

Out3

2

Out2

1

Out1Step

1/m

Gain2

k

Gain1

T

z−1

Discrete−TimeIntegrator1

T

z−1

Discrete−TimeIntegrator

1

In1

Force

2discrete oscillator force.mdl


Issues with Sprung Mass

• How well does the solution to the difference equation match

the solution to the differential equation?

• Intuitively, if T is “small enough”, approximation should be

good

• Matlab simulation of the response of the system with k = 1,

m = 1 to a step change in puck position3:

• If T is “too large”(in this case T = 0.1), response becomes

unbounded:

0 2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

3

4

time, seconds

k = 1, m = 1, period = 6.2832 seconds, T = 0.1 seconds

mas

s po

sitio

n, w

differential equationdiscrete approximation

• What goes wrong?

3Obtained with Matlab files num integration.m, oscillator.mdl, anddiscrete oscillator.mdl.


Numerical Stability

• System of differential equations:

x = Ax + Bu (2)

y = Cx + Du

• Many different ways to obtain a discrete approximation

• Simplest is the (forward) Euler scheme:

x(nT ) ≈x((n + 1)T )− x(nT )

T≈ Ax(nT )+Bu(nT )

• Difference equations:

x((n + 1)T ) = Adx(nT ) + Bdu(nT ) (3)

y(nT ) = Cdx(nT ) + Ddu(nT )

where

Ad = I + TA, Bd = TB

Cd = C, Dd = D

• Both the continuous system and the discrete appoximation are

stable if

- any bounded input results in a bounded output

- the response to initial conditions decays to zero

• Question: Does stability of (2) imply stability of (3)?


Numerical Stability, II

• The difference equation

x(n + 1) = (I + TA)x(n) + TBu(n)

is stable if the eigenvalues of the matrix I + TA have

magnitude less than one.

• The discrete transfer function G(z) = N(z)/D(z) is stable

if all the roots of D(z) = 0 lie within the unit circle

• In our case, eigenvalues4 are at λ = 1± jTp

k/m

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

real

imag

inar

y

eigenvalues at 1 ± j0.1

• If Tp

k/m is too large, the algorithm will diverge rapidly

• Other (more complicated) techniques for numerical integration

do preserve stability – but does it matter?

4num integration.m


Discrete Approximation to an Integrator

• Consider the continuous time simulation of the harmonic

oscillator in Figure 1:

wwdotwddotz

4

Force3

Puck Position

2

Velocity

1

Virtual Mass Position

k

spring constant

Step

1s

Integrator1

1s

Integrator

1/m

1/virtual mass

Figure 1: Analog Harmonic Oscillator

• The forward Euler approximation of the harmonic oscillator in

Figure 2 is obtained by replacing each analog integrator by a

discrete approximation, 1/s → T/(z − 1):

w

wdot

wddotz

3

Out3

2

Out2

1

Out1Step

1/m

Gain2

k

Gain1

T

z−1

Discrete−TimeIntegrator1

T

z−1

Discrete−TimeIntegrator

1

In1

Force

Figure 2: Discrete Harmonic Oscillator, Forward Euler

• Other discrete approximations to the harmonic oscillator may

be obtained by replacing the analog integrators by other

discrete approximations.


Discrete Approximation to an Integrator, II

Suppose we wish to approximate an integrator,

y(t) =

Z t

0

u(t)dt

by a discrete sequence y(0), y(T ), y(2T ), . . .

• Forward Euler:

y((k + 1)T ) = y(kT ) + Tu(kT )

1

s→

T

(z − 1)

T 2T 3T 4T

u(t)

t


Discrete Approximation to an Integrator, II

• Backward Euler:

y((k + 1)T ) = y(kT ) + Tu((k + 1)T )

1

s→

Tz

(z − 1)

T 2T 3T 4T

u(t)

t

• Trapezoidal:

y((k + 1)T ) = y(kT ) +T

2(u(kT ) + u((k + 1)T ))

1

s→

T (z + 1)

2(z − 1)

T 2T 3T 4T

u(t)

t


Signal Differentiation

• Used to obtain a velocity estimate from position measurements

• Problem: Differentiation amplifies noise. Consider

y(t) = sin(t)| {z }signal

+ 0.1 sin(10t)| {z }noise

• The high frequency noise is amplified5:

0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time, seconds

y(t) = sin(t) + 0.1sin(10t)

y dy/dt

• A problem even with analog differentiation! The transfer

function of a differentiator is s, and thus the frequency

response is unbounded at high frequencies!

• A pure differentiator is never implemented. Instead, an

approximate differentiator, s/(τs + 1), is used to lowpass

filter any noise

5Matlab m-file diff noise.m


Numerical Differentiation

• A digital approximation to velocity can be obtained by

approximating the derivative as

x(nT ) ≈x(nT )− x((n− 1)T )

T

• Hence, we can define a sequence of approximations ot the

derivative:

v(nT ) =x(nT )− x((n− 1)T )

T

• This procedure will still tend to amplify noise, which may still

need to be removed with a filter:

- analog filter (e.g., op amp circuit)

- digital FIR filter (e.g., moving average filter)

- digital IIR filter (e.g., lowpass filter)


Numerical Integration

• How to obtain a position estimate from velocity measurements?

• Assume that the velocity over a time interval is the average of

the velocity at the endpoints of the interval:

x(nT ) = x((n− 1)T ) +T (v(nT ) + v((n− 1)T ))

2

• Issues:

- small step size needed for rapidly varying signals, but can

cause quantization error to accumulate for slowly varying

signals

- long step size better for slowly varying signals, but can

“miss” high frequency variations.


References

[1] B. Gillespie. Tutorial on virtual environments for haptic

display. University of Michigan, April 2000.

[2] D. Sorid and S. K. Moore. The virtual surgeon. IEEE

Spectrum, pages 26–31, July 2000.


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